1.3.4 Elliptic PDE’s (Laplace, Poisson, Helmholtz)

Table 1.9: Elliptic PDE’s (Laplace, Poisson, Helmholtz) breakdown of results. Time in seconds

#	PDE	description	Mathematica		Maple		hand solved?	Animated?
			result	time	result	time
282	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge not	✓	1.78	✓	79.735	Yes
283	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, right edge not	✓	1.83	✓	8.797	Yes
284	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge has impulse	✓	4.12	✓	26.447	Yes
285	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (a)	✓	0.988	✓	72.504
286	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (b)	✓	0.944	✓	83.332
287	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (c)	✓	4.685	✓	58.768
288	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (d)	✓	4.133	✓	83.677
289	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (e)	✓	19.674	✓	73.192	Yes
290	Laplace in 2D Cartesian coordinates	Unit triangle B.C.	✓	0.985	✓	18.79
291	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	3.586	✓	7.27	Yes
292	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	2.813	✓	9.875	Yes
293	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	3.514	✓	7.235	Yes
294	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	41.351	✓	41.03	Yes
295	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	2.994	✓	32.109
296	Laplace in 2D Cartesian coordinates	Laplace PDE in 2D Cartessian with boundary condition as Dirac function	✓	0.044	✓	3.511
297	Laplace in 2D Cartesian coordinates	One side homogeneous	✓	1.692	✓	51.323
298	Laplace in 2D Cartesian coordinates	In right half plane	✓	21.834	✓	4.04
299	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	2.434	✓	23.138
300	Laplace in 2D Cartesian coordinates	Dirichlet problem Upper half	✓	1.371	✓	6.671
301	Laplace in 2D Cartesian coordinates	Right half-plane	✓	19.623	✓	4.481
302	Laplace in 2D Cartesian coordinates	First quadrant	✓	14.632	✗	31.473
303	Laplace in 2D Cartesian coordinates	Neumann problem upper half-plane	✓	4.77	✓ used convert(sol,Int).	8.997
304	Laplace in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	2.303	✓	12.098
305	Laplace in 2D Cartesian coordinates	Strip in upper half	✓	60.258	✓	12.959
306	Laplace in 2D Cartesian coordinates	in Rectangle, right edge at infinity	✗	2.353	✓	19.588
307	Laplace in 2D Polar coordinates	Laplace PDE inside quarter disk, Neumann BC at edge	✓	3.459	✓	2.842
308	Laplace in 2D Polar coordinates	\(r=4\) and \(u=x^4\) at boundary of disk	✓	150.086	✓	22.021	Yes
309	Laplace in 2D Polar coordinates	\(r=1\) and \(u_r=x\) at boundary of disk	✗	2.589	✗	1.888	Yes
310	Laplace in 2D Polar coordinates	Laplace inside disk. General solution	✓	0.624	✓	11.463
311	Laplace in 2D Polar coordinates	Laplace inside disk. Specific boundary conditions	✓	3.936	✓	13.769	Yes
312	Laplace in 2D Polar coordinates	Haberman 2.5.5 (c)	✗	3.989	✓	2.611	Yes
313	Laplace in 2D Polar coordinates	semi-circle	✓	2.08	✓	2.322
314	Laplace in 2D Polar coordinates	Haberman 2.5.8 (b)	✓	8.91	✓	361.819	Yes
315	Laplace in 2D Polar coordinates	Circular annulus	✓	0.519	✓	9.499
316	Laplace in 2D Polar coordinates	Outside a disk	✓	6.915	✓	13.615
317	Laplace in 2D Polar coordinates	Outside a disk	✓	4.21	✓	15.946	Yes
318	Laplace in 3D Spherical coordinates	In a sphere	✓	0.036	✓	3.401
319	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (a)	✗	0.018	✗	3.513
320	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (b)	✗	0.015	✗	3.773
321	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (c)	✗	0.015	✗	3.813
322	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (d)	✗	0.014	✗	3.881
323	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (e)	✗	0.015	✗	3.827
324	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (a)	✗	0.015	✗	4.865
325	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (b)	✗	0.015	✗	39.671
326	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (c)	✗	0.02	✗	53.644
327	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (d)	✗	0.014	✗	4.81
328	Poisson in 2D Cartesian coordinates	All boundaries at zero	✗	0.015	✗	82.75	Yes
329	Poisson in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	1.368	✓	1.599
330	Poisson in 2D Cartesian coordinates	Poisson PDE in whole 2D plane	✓	31.393	✓	0.066	Yes
331	Helmholtz in 2D Cartesian coordinates	In rectangle	✓	64.18	✓	25.968
332	Helmholtz in 2D Cartesian coordinates	On whole plane	✓ why? It solved earlier with BC?	0.004	✓	0.289
333	Helmholtz in 2D Cartesian coordinates	Reduced Helmholtz Inside square	✓	62.146	✓	26.367
334	Helmholtz in 3D Spherical coordinates	Chain reaction PDE	✗	0.123	✗ trivial solution	4.683	Yes